How would you use Project Management to add value to your Customer?

Structure and professionalism

George, what domains are your project involving? A little more detail would be helpful too since all of our advice, suggestions and wisdom can focus better. My companies for which I created or handled projects, had not really used what we now call PMBOK or BABOK guidelines. SMEs are essential to success and sometimes they have never been used by the customer. But then is the customer hiring you personally or your PMO? Customers need management and information. So that is critical to your own successful completion of the project or each of its life cycles. I created an Access Database twice in two different companies to help track not so much the Project Progress, but the vendor-consumer-customer feedback. I tied it to all current products, and the new ones being created, and entered EVERY Phone call and date and time, and a major list of all issues and questions, with domains involved and folloups, everything!
This allowed me everyday or week, or bi-monthly to provide a log and report in all different ways and then present that to stakeholders at the meetings.
This IMPROVED Communication Management, Change Control, Risk and Scope monitoring. We found that previous Requirements and Assumptions and needs were off and stakeholders misread the needs or forgot some!, and had to adjust scope. I finished the project a little ahead of time but minor over-budget (I had reserves) , but it succeeded.

Adding value to your Customer is a very general statement as we dont know what customer it is, ie consumer, normal customer contracts like a die-maker or IT ?

For me, adding value to a customer means ensuring that at all times, the project is still aligned to the business objectives for implementing the project and also ensuring that the end product meets or exceeds the customer expectations and requirements. Yes, it's the Project Manager's responsibility to manage scope, costs, relationship, quality, etc. But at the end, the value will be realised if the business objectives are still aligned to and the project delivered meets the client needs.

One of the pre-requisites would be successful completion of respective project under customer's strategic program.
Post that to add value to the program and to customer as well. This may not necessary mean alignment under respective process groups in each project, however it needs that strategically and financially where customer wants to reach (per roadmap) and guide the customer on your challenges, lessons learns and consultation over what needs to be done next (strictly in terms of functional department of the customer and impacting project deliverables) for advantage in the market.

All comments above are valid. Business has an idea - creates a business case usually based on generalities. By the Time the PM is involved the project is approved with a set budget, time boxed and provided very limited resources.
The PM adds values as the business partner and technology partner, working with and through others to solve the problems, setting expectations. Bridging the gap between technology and users. Enabling user efficiencies and saving the organization monies. Transparent communications for informed decision, removing impediments, maintaining the big picture, ensuring tasks are completed as scheduled meeting expectations and saving monies. We add value by Problem solving. Collaboration, providing resolution, coordination and organizing activates to achieve desired results.